International Journal of Geometric Methods in Modern Physics

The apparent contour of a surface in $3$-space can be investigated in terms of singularity theory. We show the precise bifurcation diagrams of the apparent contours of generic crosscap surfaces with respect to orthogonal projections. Especially, our bifurcation diagrams contain also the information of the projected images of the singular sets of crosscap surfaces.

We develop further the general framework for modeling the dynamics of the motion of black holes, presented in [E. Gallo and O. M. Moreschi, Modeling the dynamics of black holes through balanced equations of motion, Int. J. Geom. Meth. Mod. Phys. 16(3) (2019) 1950034], by employing a convenient null gauge, in general relativity, for the construction of the balanced equations of motion. This null gauge has the property that the asymptotic structure is intimately related to the interior one; in particular there is a strong connection between the field equations and the balanced equations of motion.

Our work is very related to what we have called “Robinson–Trautman (RT) geometries” [S. Dain, O. M. Moreschi and R. J. Gleiser, Photon rockets and the Robinson–Trautman geometries, Class. Quantum Grav. 13(5) (1996) 1155–1160] in the past. These geometries are used in the sense of the general framework, we have presented in [E. Gallo and O. M. Moreschi, Modeling the dynamics of black holes through balanced equations of motion, Int. J. Geom. Meth. Mod. Phys. 16(3) (2019) 1950034].

We present the balanced equations of motion in second order of the acceleration. We solve the required components of the field equation at their respective required orders, $G^2$ and $G^3$, in terms of the gravitational constant.

We indicate how this approach can be extended to higher orders.

The aim of this paper is to study the charged anisotropic strange stars in the Rastall framework. Basic formulation of field equations in this framework is presented in the presence of charged anisotropic source. To obtain the solutions of the Rastall field equations in spherically symmetric Karori and Barua (KB) type space-time, we have considered a linear equation of state of strange matter, using the MIT bag model. The constraints on the Rastall dimensionless parameter $\gamma$ are also discussed to obtain the physically reasonable solution. We explore some physical features of the presented model like energy conditions, stability and hydrostatic equilibrium, which are necessary to check the physical viability of the model. We also sought for the influence of the Rastall dimensionless parameter on the behavior of the physical features of obtained solution. We plot the graphs of matter variables for different chosen values of the parameter $\gamma$ to inspect more details of analytical investigations and predict the numerical values of these variables exhibited in the tabular form. For this analysis, we choose four different arbitrary models of strange stars with compactness $u(=\frac{M}{R})$ 0.25, 0.30, 0.35 and 0.40. We observed that all the necessary physical conditions are satisfied and the presented model is quite reasonable to study the strange stars.

In a $n$-dimensional Friedmann–Robertson–Walker metric, it is rigorously shown that any analytical theory of gravity $f(R,{𝒢})$, where $R$ is the curvature scalar and ${𝒢}$ is the Gauss–Bonnet topological invariant, can be associated to a perfect-fluid stress–energy tensor. In this perspective, dark components of the cosmological Hubble flow can be geometrically interpreted.

In this paper, we study para-Sasakian manifold $(M,g)$ whose metric $g$ is an $\eta$-Ricci soliton $(g,V)$ and almost $\eta$-Ricci soliton. We prove that, if $g$ is an $\eta$-Ricci soliton, then either $M$ is Einstein and in such a case the soliton is expanding with $\lambda =2n$ or it is $D$-homothetically fixed $\eta$-Einstein manifold and in such a case the soliton is shrinking with $\lambda =-2n-4$. We show the same conclusion when the para-Sasakian manifold $(M,g)$ is of $\mathrm{\text{dim}}>3$ and $g$ is an almost $\eta$-Ricci soliton with $V$ as infinitesimal contact transformation. Finally, we prove that, if the para-Sasakian manifold $(M,g)$ of $\mathrm{\text{dim}}>3$ admits a gradient almost $\eta$-Ricci soliton with $V\ne 0$, then $M$ is Einstein. Suitable examples are constructed to justify our results.

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

The kinematical foundations of Schwinger’s algebra of selective measurements were discussed in [F. M. Ciaglia, A. Ibort and G. Marmo, Schwinger’s picture of quantum mechanics I: Groupoids, To appear in IJGMMP (2019)] and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this paper, the dynamical aspects of the theory are analyzed. For that, the algebra generated by the observables, as well as the notion of state, are discussed, and the structure of the transition functions, that plays an instrumental role in Schwinger’s picture, is elucidated. A Hamiltonian picture of dynamical evolution emerges naturally, and the formalism offers a simple way to discuss the quantum-to-classical transition. Some basic examples, the qubit and the harmonic oscillator, are examined, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures is discussed.

In this paper, it is shown that the reversible Blaschke Finsler metrics on spheres with vanishing $S$-curvature are Riemannian.

Using the toroidal compactification of string theory on $n$-dimensional tori, ${𝕋}^n$, we investigate dyonic objects in arbitrary dimensions. First, we present a class of dyonic black solutions formed by two different D-branes using a correspondence between toroidal cycles and objects possessing both magnetic and electric charges, belonging to $\text{U}{(1)}_e^{2^{n-1}}\times \text{U}{(1)}_m^{2^{n-1}}$ dyonic gauge symmetry. This symmetry could be associated with electrically charged magnetic monopole solutions in stringy model buildings of the standard model (SM) extensions. Then, we consider in some detail such black hole classes obtained from even-dimensional toroidal compactifications, and we find that they are linked to $Cl(n)$ Clifford algebras using the vee product. It is believed that this analysis could be extended to dyonic objects which can be obtained from local Calabi–Yau manifold compactifications.

The purpose of this paper is to discuss the theoretical study of a nonlinear problem of cilia induced flow by considering the fluid as anincompressible non-Newtonian fluid (hyperbolic tangent fluid) model by means of ciliated walls. The leading equations of present flow problem are simplified under the consideration of long-wavelength approximation. We have utilized regular perturbation technique to solve the simplified leading equations of hyperbolic tangent fluid model. The analytical solution is computed for stream function and numerical solution is computed for the rise in pressure. The characteristics of the ciliary system on tangent hyperbolic fluid are analyzed graphically and discussed in detail. It has been found that when $n=0$, the results of pressure rise coincide with the results of Newtonian fluid. It has also been observed that the size of the trapping bolus decreases with an increase in Hartmann number and Weissenberg number.

In this paper, we study the Dirac equation in the Rindler spacetime. The solution of the wave equation in an accelerated reference frame is obtained. The differential equation associated to this wave equation is mapped into a Sturm–Liouville problem of a Schrödinger-like equation. We derive a compact expression for the energy spectrum associated with the Dirac equation in an accelerated reference. It is shown that the noninertial effect of the accelerated reference frame mimics an external potential in the Dirac equation and, moreover, allows the formation of bound states.

In this paper, we study viscous Modified Cosmic Chaplygin Gas (MCCG) in the presence of cosmological constant in flat FRW universe. We assume that bulk viscosity $\zeta$ and cosmological constant $\mathrm{\Lambda }$ are the linear combinations of two terms, one is constant and other is a function of dark energy density $\rho$. In this framework, we solve the non-linear differential equation analytically and numerically and obtain time dependent dark energy density. We also consider two separate cases of early and late universe and discussed the evolution of dark energy density. We investigate the effect of viscosity and cosmological constant to the evolution of universe and discuss the stability of the model by square of speed of sound. Finally, we compare our model with Cardassian universe.

The upper bound of Ricci curvature conjecture, also known as Chen-Ricci conjecture, was formulated by Chen [B. Y. Chen, Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension, Glasgow Math. J. 41 (1999) 33–41] and modified by Tripathi [M. M. Tripathi, Improved Chen–Ricci inequality for curvature-like tensors and its applications, Diff. Geom. Appl. 29 (2011) 685–698]. In this paper, first, we define partially minimal isometric immersion of warped product manifolds. Then, we derive a fundamental theorem for Ricci curvature via partially minimal isometric immersions from a warped product pointwise bi-slant submanifolds into complex space forms. Some applications are constructed in terms of Dirichlet energy function, Hamiltonian, Lagrangian and Hessian tensor due to appearance of the positive differential function in the inequality.

This paper investigates the possibility of static spherical symmetric wormholes existence exhibiting conformal motion in a generalized teleparallel formulation, namely $f(T)$ gravity. For this purpose, we assume the matter distribution as anisotropic fluid with energy density of Gaussian distribution, a well-known non-commutative aspect of string theory. By using non-diagonal tetrad components in the torsional formulation, we consider three interesting viable models of $f(T)$. In each case, due to the complexity of the resulting field equations, it is seen that the analytic solution is difficult to find. Thus the feasible and realistic wormholes are acquired using numerical approach by fixing the involved free parameters suitably. Furthermore, we investigate the stability of these wormhole solutions graphically and it is shown that the obtained wormhole solutions are physically stable in all cases.

We discuss the thermodynamical behavior of homogeneous and isotropic universe (flat and non-flat) in the framework of $f(R,T^{\varphi })$ gravity, where $R$ stands for Ricci scalar and $T^{\varphi }$ signifies the trace of energy–momentum tensor of a scalar field $\varphi$. We follow through the first-order formalism that specifies the scalar field to the Hubble parameter which becomes $H=W(\varphi ).$ By using Bekenstein–Hawking entropy, we analyze the validity of generalized second law of thermodynamics at apparent horizon for different values of $W(\varphi )$ and evaluate the equilibrium condition for all cases as well.

The paper is devoted to 2-spinor calculus methods in general relativity. New spinor form of the Lovelock differential identity is suggested. This identity is second-order identity for the Riemann curvature tensor. We provide an example that our spinorial treatment of Lovelock identity is effective for the description of solutions of Einstein–Maxwell equations. It is shown that the covariant divergence of Lipkin’s zilch tensor for the free Maxwell field vanishes on the solutions of Einstein–Maxwell equations if and only if the energy–momentum tensor of the electromagnetic field is Weyl-compatible.

In the most general geometric background, we study the Dirac spinor fields with particular emphasis given to the explicit form of their gauge momentum and the way in which this can be inverted so as to give the expression of the corresponding velocity; we study how Zitterbewegung affects the motion of particles, focusing on the internal dynamics involving the chiral parts; we discuss the connections to field quantization, sketching in what way anomalous terms may be gotten eventually.